1. Functions 2

2. Chapter Overview Perhaps the most useful mathematical idea for modeling the real world is the concept of function, which we study in this chapter. To understand what a function is, let’s look at an example.

3. Chapter Overview If a rock climber drops a stone from a high cliff, what happens to the stone?

4. Chapter Overview Of course the stone falls. How far it has fallen at any given moment depends upon how long it has been falling. That’s a general description. However, it doesn’t tell us exactly when the stone will hit the ground.

5. Chapter Overview What we need is a rule that relates the position of the stone to the time it has fallen. Physicists know that the rule is: In t seconds, the stone falls 16t 2 feet.

6. Chapter Overview If we let d(t) stand for the distance the stone has fallen at time t, we can express this rule as: d(t) = 16t 2

7. Chapter Overview This “rule” for finding the distance in terms of the time is called a function. We say that distance is a function of time.

8. Chapter Overview To understand this rule or function better, we can make a table of values or draw a graph.

9. Chapter Overview The graph allows us to easily visualize how far and how fast the stone falls.

10. Chapter Overview You can see why functions are important. If a physicist finds the “rule” or function that relates distance fallen to elapsed time, she can predict when a missile will hit the ground.

11. Chapter Overview If a biologist finds the function or “rule” that relates the number of bacteria in a culture to the time, he can predict the number of bacteria for some future time. If a farmer knows the function or “rule” that relates the yield of apples to the number of trees per acre, he can decide how many trees per acre to plant to maximize the yield.

12. Chapter Overview In this chapter, we will learn: How functions are used to model real-world situations. How to find such functions.

13. What Is a Function? 2.1

14. What Is a Function? In this section, we: Explore the idea of a function. Give the mathematical definition of function.

15. Functions All Around Us

16. In nearly every physical phenomenon, we observe that one quantity depends on another.

17. Functions All Around Us For example, Your height depends on your age. The temperature depends on the date. The cost of mailing a package depends on its weight.

18. Functions All Around Us We use the term function to describe this dependence of one quantity on another. That is, we say: Height is a function of age. Temperature is a function of date. Cost of mailing a package is a function of weight.

19. Functions All Around Us The U.S. Post Office uses a simple rule to determine the cost of mailing a package based on its weight. However, it’s not so easy to describe the rule that relates height to age or temperature to date.

20. Functions All Around Us Here are some more examples: The area of a circle is a function of its radius. The number of bacteria in a culture is a function of time. The weight of an astronaut is a function of her elevation. The price of a commodity is a function of the demand for that commodity. Temperature of water from the faucet is a function of time.

21. Area of a Circle The rule that describes how the area A of a circle depends on its radius r is given by the formula A = πr 2

22. Temperature of Water from a Faucet Even when a precise rule or formula describing a function is not available, we can still describe the function by a graph. For example, when you turn on a hot water faucet, the temperature of the water depends on how long the water has been running. So we can say that: Temperature of water from the faucet is a function of time.

23. Temperature of Water from a Faucet The figure shows a rough graph of the temperature T of the water as a function of the time t that has elapsed since the faucet was turned on.

24. Temperature of Water from a Faucet The graph shows that the initial temperature of the water is close to room temperature. When the water from the hot water tank reaches the faucet, the water’s temperature T increases quickly.

25. Temperature of Water from a Faucet In the next phase, T is constant at the temperature of the water in the tank. When the tank is drained, T decreases to the temperature of the cold water supply.

26. Definition of a Function

27. Function A function is a rule. To talk about a function, we need to give it a name. We will use letters such as f, g, h,... to represent functions. For example, we can use the letter f to represent a rule as follows: “f”is the rule“square the number”

28. Function When we write f(2), we mean “apply the rule f to the number 2.” Applying the rule gives f(2) = 2 2 = 4. Similarly, f(3) = 3 2 = 9 f(4) = 4 2 = 16 and, in general, f(x) = x 2

29. Function—Definition A function f is: A rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

30. Function We usually consider functions for which the sets A and B are sets of real numbers. The symbol f(x) is read “f of x” or “f at x.” It is called the value of f at x, or the image of x under f.

31. Domain & Range The set A is called the domain of the function. The range of f is the set of all possible values of f(x) as x varies throughout the domain, that is, range of f = {f(x) | x T  A}

32. Independent and Dependent Variables The symbol that represents an arbitrary number in the domain of a function f is called an independent variable. The symbol that represents a number in the range of f is called a dependent variable.

33. Independent and Dependent Variables So, if we write y = f(x) then x is the independent variable. y is the dependent variable.

34. Machine Diagram It’s helpful to think of a function as a machine. If x is in the domain of the function f, then when x enters the machine, it is accepted as an input and the machine produces an output f(x), according to the rule of the function.

35. Machine Diagram Thus, we can think of: The domain as the set of all possible inputs. The range as the set of all possible outputs.

36. Arrow Diagram Another way to picture a function is by an arrow diagram. Each arrow connects an element of A to an element of B. The arrow indicates that f(x) is associated with x, f(a) is associated with a, and so on.

37. E.g. 1—The Squaring Function The squaring function assigns to each real number x its square x 2. It is defined by f(x) = x 2. (a) Evaluate f(3), f(–2), and f( ). (b) Find the domain and range of f. (c) Draw a machine diagram for f.

38. E.g. 1—The Squaring Function The values of f are found by substituting for x in f(x) = x 2. f(3) = 3 2 = 9 f(–2) = (–2) 2 = 4 f( ) = ( ) 2 = 5 Example (a)

39. E.g. 1—The Squaring Function The domain of f is the set of all real numbers. The range of f consists of all values of f(x), that is, all numbers of the form x 2. Since x 2 ≥ 0 for all real numbers x, we can see that the range of f is: {y | y ≥ 0} = [0, ∞) Example (b)

40. E.g. 1—The Squaring Function Here’s a machine diagram for the function. Example (c)

41. Evaluating a Function

42. In the definition of a function the independent variable x plays the role of a “placeholder.” For example, the function f(x) = 3x 2 + x – 5 can be thought of as: f(__) = 3 · __ 2 + __ –5 To evaluate f at a number, we substitute the number for the placeholder.

43. E.g. 2—Evaluating a Function Let f(x) = 3x 2 + x – 5. Evaluate each function value. (a) f(–2) (b) f(0) (c) f(4) (d) f(½)

44. E.g. 2—Evaluating a Function To evaluate f at a number, we substitute the number for x in the definition of f.

45. E.g. 3—A Piecewise Defined Function A cell phone plan costs $39 a month. The plan includes 400 free minutes and charges 20¢ for each additional minute of usage. The monthly charges are a function of the number of minutes used, given by: Find C(100), C(400), and C(480).

46. E.g. 3—A Piecewise Defined Function Remember that a function is a rule. Here’s how we apply the rule for this function. First, we look at the value of the input x. If 0 ≤ x ≤ 400, then the value of C(x) is: 39 However, if x > 400, then the value of C(x) is: 39 + 0.2(x – 400)

47. E.g. 3—A Piecewise Defined Function Since 100 ≤ 400, we have C(100) = 39. Since 400 ≤ 400, we have C(400) = 39. Since 480 > 400, we have C(480) = 39 + 0.2(480 – 400) = 55. Thus, the plan charges: $39 for 100 minutes, $39 for 400 minutes, and $55 for 480 minutes.

48. E.g. 4—Evaluating a Function If f(x) = 2x 2 + 3x – 1, evaluate the following.

49. E.g. 4—Evaluating a Function

50. (d) Using the results from parts (c) and (a), we have:

51. E.g. 5—The Weight of an Astronaut If an astronaut weighs 130 pounds on the surface of the earth, then her weight when she is h miles above the earth is given by the function

52. E.g. 5—The Weight of an Astronaut (a) What is her weight when she is 100 mi above the earth? (b) Construct a table of values for the function w that gives her weight at heights from 0 to 500 mi. What do you conclude from the table?

53. E.g. 5—The Weight of an Astronaut We want the value of the function w when h = 100. That is, we must calculate w(100). So, at a height of 100 mi, she weighs about 124 lb. Example (a)

54. E.g. 5—The Weight of an Astronaut The table gives the astronaut’s weight, rounded to the nearest pound, at 100-mile increments. The values are calculated as in part (a). The table indicates that, the higher the astronaut travels, the less she weighs. Example (b)

55. The Domain of a Function

56. Domain of a Function Recall that the domain of a function is the set of all inputs for the function. The domain of a function may be stated explicitly. For example, if we write f(x) = x 2, 0 ≤ x ≤ 5 then the domain is the set of all real numbers x for which 0 ≤ x ≤ 5.

57. Domain of a Function If the function is given by an algebraic expression and the domain is not stated explicitly, then, by convention, the domain of the function is: The domain of the algebraic expression—that is, the set of all real numbers for which the expression is defined as a real number.

58. Domain of a Function For example, consider the functions The function f is not defined at x = 4. So, its domain is {x | x ≠ 4 }. The function g is not defined for negative x. So, its domain is {x | x ≠ 0 }.

59. E.g. 6—Finding Domains of Functions Find the domain of each function.

60. E.g. 6—Finding Domains The function is not defined when the denominator is 0. Since we see that f(x) is not defined when x = 0 or x = 1. Example (a)

61. E.g. 6—Finding Domains Thus, the domain of f is: {x | x ≠ 0, x ≠ 1} The domain may also be written in interval notation as: (∞, 0)  (0, 1)  (1, ∞) Example (a)

62. E.g. 6—Finding Domains We can’t take the square root of a negative number. So, we must have 9 – x 2 ≥ 0. Using the methods of Section 1-7, we can solve this inequality to find that: –3 ≤ x ≤ 3 Thus, the domain of g is: {x | –3 ≤ x ≤ 3} = [–3, 3] Example (b)

63. E.g. 6—Finding Domains We can’t take the square root of a negative number, and we can’t divide by 0. So, we must have t + 1 > 0, that is, t > –1. Thus, the domain of h is: {t | t > –1} = (–1, ∞) Example (c)

64. Four Ways to Represent a Function

65. To help us understand what a function is, we have used: Machine diagram Arrow diagram

66. Four Ways to Represent a Function We can describe a specific function in these ways: Verbally (a description in words) Algebraically (an explicit formula) Visually (a graph) Numerically (a table of values)

67. Four Ways to Represent a Function A single function may be represented in all four ways. It is often useful to go from one representation to another to gain insight into the function. However, certain functions are described more naturally by one method than by the others.

68. Verbal Representation An example of a verbal description is: P(t) is “the population of the world at time t”

69. Numerical Representation The function P can also be described numerically by giving a table of values.

70. Algebraic Representation A useful representation of the area of a circle as a function of its radius is the algebraic formula A(r) 2 = πr 2

71. Visual Representation The graph produced by a seismograph is a visual representation of the vertical acceleration function a(t) of the ground during an earthquake.

72. Verbal Representation Finally, consider the function C(w). It is described verbally as: “the cost of mailing a first-class letter with weight w.”

73. Verbal/Numerical Representation The most convenient way of describing this function is numerically—using a table of values.

74. Four Ways to Represent a Function We will be using all four representations of functions throughout this book. We summarize them in the following box.

75. Four Ways to Represent a Function